Parallel and perpendicular lines
Parallel lines
lines have the . They never meet.
To find a parallel line through a point:
- Use the same gradient m as the line you're given.
- Substitute the point into y = mx + c and find c.
Example: find the equation of the line parallel to y = 2x + 5 that passes through (3, 1).
- m = 2
- 1 = 2 × 3 + c, so c = −5
- y = 2x − 5
Example: are y = 3x + 1 and 2y = 6x − 4 parallel?
- Rearrange the second: y = 3x − 2
- Both have gradient 3, so yes
Key terms
Perpendicular lines (Higher tier only)
lines meet at a right angle. Their gradients multiply to give −1, so each is the of the other: flip it and change its sign.
| Gradient | Perpendicular gradient |
|---|---|
| 2 | −½ |
| −3 | ⅓ |
| 2/3 | −3/2 |
Example: find the equation of the line perpendicular to y = (2/3)x + 1 that passes through (4, 1).
- Perpendicular gradient: −3/2, or −1.5
- 1 = −1.5 × 4 + c, so 1 = −6 + c
- c = 7, so y = −1.5x + 7
Check: 2/3 × −3/2 = −1.
Common mistake
Do both steps: flip and change the sign. The perpendicular gradient to 2 is −½, not ½ or −2.
Key terms
Perpendicular bisectors (Higher tier only)
The of AB crosses it at its midpoint, at a right angle.
- Find the midpoint of AB.
- Find the gradient of AB, then its negative reciprocal.
- Substitute the midpoint into y = mx + c to find c.
Example: A is (1, 2) and B is (5, 10). Find the equation of the perpendicular bisector of AB.
- Midpoint: ((1 + 5) ÷ 2, (2 + 10) ÷ 2) = (3, 6)
- Gradient of AB = (10 − 2) ÷ (5 − 1) = 8 ÷ 4 = 2, so the perpendicular gradient is −½
- 6 = −½ × 3 + c, so c = 7.5
- y = −½x + 7.5
Check: the midpoint is on the line, since −½ × 3 + 7.5 = 6.